TECHNICAL COMMENT
Comment on ‘‘The Illusion of Invariant
Quantities in Life Histories’’
Van M. Savage,1* Ethan P. White,2,4 Melanie E. Moses,3 S. K. Morgan Ernest,2
Brian J. Enquist,4 Eric L. Charnov3
Nee et al. (Reports, 19 August 2005, p. 1236) used a null model to argue that life history
invariants are illusions. We show that their results are largely inconsequential for life history theory
because the authors confound two definitions of invariance, and rigorous analysis of their null
model demonstrates that it does not match observed data.
ecades of research on life history invariants have identified deep symmetries in evolutionary biology that reveal
fundamental and pervasive constraints upon
diverse organisms (1, 2). Nee et al. (3) constructed null models to argue that slopes and R2
from log-log plots cannot reliably identify invariants. This analysis has led others to conclude that life history invariants may not exist
(4, 5). These conclusions are erroneous because
they are based on a misrepresentation of life
history invariants and on a failure to recognize that properties of these null models differ significantly from empirical data. Here,
we show that Nee et al._s results are largely
inconsequential for life history and allometric
theories.
Within life history theory, the term invariance is used in two ways. In type A invariance, a biological characteristic does not
vary systematically with another characteristic,
such as body size (6); in type B invariance, a
biological characteristic exhibits a unimodal
central tendency and varies over a limited range
(7, 8). For example, the ratio of weaning mass
to adult mass in mammals is a type A invariant
because its value shows no trend with body
size. In addition, this ratio is a type B invariant
because weaning mass is typically close to 30%
of adult mass. In his original work, Charnov (2)
examines both types of invariants; however, he
clearly emphasizes type A as the more important.
He begins his book by stating that BSomething
will be called invariant (or an invariant) if it does
not change under the specified transformation[
and Bthe underlying transformation is adult
body size between species.[
The null models of Nee et al., although presented and interpreted as posing problems for all
life history invariants, are only relevant for de-
D
1
Bauer Center for Genomics Research, Harvard University,
Cambridge, MA 02138, USA. 2Department of Biology, Utah
State University, Logan, UT 84322, USA. 3Department of
Biology, University of New Mexico, Albuquerque, NM
87131, USA. 4Department of Ecology and Evolutionary
Biology, University of Arizona, Tucson, AZ 85721, USA.
*To whom correspondence should be addressed. E-mail:
vsavage@cgr.harvard.edu
tecting type B invariance, as we now explain. If
(i) the ratio of life history characteristics, c 0 y/x,
is randomly distributed and (ii) VarEln(x)^ d
VarEln(c)^, then it is straightforward to show
analytically that the slope and R2 of ln( y) versus
ln(x) are near 1. Nee et al._s null models satisfy
these assumptions (3). By assuming condition (i),
that y/x does not vary systematically with another
variable, the simulated data of Nee et al. are type
A invariants, by definition. Thus, they do not
provide an alternative null model to this type of
invariance. Indeed, the Nee et al. results demonstrate that slopes and R2 near 1 from log-log plots
are valid for identifying type A invariance. Moreover, when c is drawn from a uniform random
distribution with any reasonable choice of bounds
(9), condition (ii) is satisfied, and therefore R2 ,
1. Thus, Nee et al._s choice of the uniform
random distribution, but not the specific bounds,
is crucial for obtaining their results and, as such,
the critical comparison to determine whether the
observed data are described by this null is to
compare the distribution of invariants to a uniform random distribution (10).
When the null results of Nee et al. are rigorously compared with existing empirical data,
Fig. 1. Plot of age at sexual maturity versus
average adult life span for several taxa (13). The
slope is the value of the life history ratio, and in
all cases except the reptiles, the values differ
significantly from those predicted by Nee et al.’s
null (P G 0.01 in all cases; based on 10,000
draws of a mean value from the appropriate sample size).
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it becomes obvious that the null fails to predict important biological properties of life
history invariants and thus fails to describe
type B invariance as well. First, the intercept
from a regression on their null predicts a life
history ratio, given by the distribution mean,
that is independent of taxa. Therefore, Nee
et al. cannot account for the observation that
different clades or taxa are described by different values of c, representing important evolutionary differences between taxonomic groups
(Fig. 1) (2). Second, many observed distributions of life history invariants are unimodal
with a constrained range and thus significantly
differ from Nee et al._s null model (Fig. 2).
Third, their null model predicts R2 values
that are in fact lower than those for real data
(Fig. 2) (11, 12). Fourth, Nee et al._s testing of
Fig. 2. Plots of observed data (in this case
weaning mass/adult mass) and of Nee et al.’s null
model based on the observed adult mass distribution. The observed data exhibit both types of
invariants. Both real [77 values for mammals
(13)] and simulated values of w/m are invariant
with respect to size (P 9 0.1; lines are means),
demonstrating that Nee et al.’s null assumes type
A invariance. In addition to showing no trend with
mass, histograms of invariants frequently differ
significantly from Nee et al.’s uniform random
distribution. For example, the distribution of w/m
has a clear mode, contrary to the predictions of
Nee et al.’s model (K-S test for deviation from
uniform: P G 10j3). Evidence for this type of
deviation is present in the literature in the form
of plots of the distributions of invariants and
linear plots of the relevant variables (2). In addition, their null predicts R2 values that, while
high, are often significantly lower than those for
real data (for this example, P G 10–5; based on
100,000 randomizations combining the observed
adult mass distribution with 77 draws from a
uniform distribution).
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TECHNICAL COMMENT
the ratio of annual clutch size to annual mortality rate against their null is clearly incorrect.
Instead of using their null to calculate an
expected R2, they use Ricklefs_ (13) empirical
data to calculate the R 2 of the data, which is a
statistical fact unrelated to their null model.
Nee et al. incorrectly cite this as evidence
against life history invariants. Using a uniform
random distribution, corresponding to Nee
et al._s null model, we find R2 0 0.44, much
lower than that of Ricklefs_ empirical data
(R2 0 0.84) and thus, evidence against Nee
et al._s null.
Further, contrary to claims by Nee et al. and
others that life history theory implies that invariants show no variation, Charnov Ee.g., pp. 5
and 15 of (2)^ clearly states that a distribution
is expected for any life history invariant. Indeed,
life history theory endeavors to understand how
natural selection sets both the central tendency
and the distribution of these ratios (2, 7, 11).
Finally, Nee et al._s null model can only produce
slopes near 1 and, thus, contrary to concerns
raised by de Jong and others (4, 5), cannot explain the ubiquitous quarter-power slopes observed in allometry or, by extension, the life
history invariants formed from them (1, 2, 7).
Life history invariants are governed by nonrandom processes and form the cornerstone
of a general framework that mechanistically
links variation in organismal form, function,
ecology, and evolution across differing environments Ee.g., (1, 2, 8)^. For more than 40 years,
fishery science has used them with great
success Ee.g., (14)^. Life history invariants are
certainly not illusions.
198b
References and Notes
1. W. A. Calder, Size, Function, and Life History (Harvard
Univ. Press, Cambridge, MA, 1984).
2. E. L. Charnov, Life History Invariants: Some Explorations
of Symmetry in Evolutionary Ecology (Oxford Univ. Press,
Oxford, 1993).
3. S. Nee, N. Colgrave, S. A. West, A. Grafen, Science 309,
1236 (2005).
4. G. de Jong, Science 309, 1193 (2005).
5. Faculty of 1000 Biology: Evaluations for Nee S et al.
Science 2005 Aug 19 309 (5738):1236-9;
www.f1000biology.com/article/16109879/evaluation.
6. The fact that certain quantities are invariant with respect
to a given variable or transformation is the traditional
definition of invariant in math and statistics and also the
one clearly given in regards to life history theory (2). This
is because the vast majority of life history parameters
depend on body mass [e.g., (1, 2)], so that invariants
represent an important and meaningful exception to the
rule (2). For example, the number of heartbeats in an
organism’s lifetime does not change systematically as a
function of body size.
7. This use of invariant (addressed by Nee et al.) is not well
defined in the literature. Our reading of it is based on
statements such as, ‘‘This [invariant] suggests that there
is a fundamental similarity across all animals, from a
2-mm-long crustacean to a 1.5-m-long fish, in the
underlying forces that select for sex change’’ [(8); see
also (2)]. This implies a unimodal distribution, suggestive of
an optimal value. That this value may be optimal is further
revealed by examination of the corresponding allometric
relationships. For example, primates have slow biomass
production rates, which distinguishes them from other
mammals when life history characteristics are plotted
against mass [figures 1.13 and 6.4 in (2)]. However,
primates are similar to other mammals when dimensionless
comparisons are made (e.g., yearly clutch size multiplied by
age at maturity is invariant [figure 1.3 in (2)] because
biomass production rate cancels out in this ratio.
8. D. J. Allsop, S. A. West, Nature 425, 783 (2003).
9. If c is drawn from a uniform random distribution with a
lower bound of 0, then Var[ln(c)] 0 1, condition (ii) is
satisfied for most data, and R2 , 1, regardless of the upper
bound for c. If the upper bound is less than a billion, then
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10.
11.
12.
13.
14.
15.
Var[ln(c)] G 100, condition (ii) is satisfied for most data,
and R2 , 1, regardless of the lower bound for c. Thus,
although Nee et al.’s choice of uniform random distributions is crucial for obtaining their results, the specific
choice of bounds (0 e c e1), which they and others
emphasize as important (4, 5), is essentially irrelevant. This
is crucial because otherwise their Eq. 4 would be invalid as
E is clearly not bounded between 0 and 1.
Contrary to Nee et al.’s suggestion that distributions
other than the uniform random could be used for
generating values for c, it is clear that the uniform
random (constrained by logical bounds) is the only
distribution justified as a true null based on the
bounding of the data (12). Any other distribution would
have the potential to smuggle biology into the null model.
A. Gardner, D. J. Allsop, E. L. Charnov, S. A. West, Am.
Nat. 165, 551 (2005).
R. Cipriani, R. Collin, J. Evol. Biol. 18, 1613 (2005).
Methods are available as supporting material on
Science Online.
R. J. H. Beverton, S. J. Holt, in Ciba Foundation Colloquia
in Ageing. V. The Lifespan of Animals, G. E. W.
Wolstenholme, M. O’Connor, Eds. (Churchill, London,
1959), pp. 142–177.
We thank A. Allen, J. Brown, J. Gillooly, and G. West for
helpful discussions during the development of this
comment. V.M.S. acknowledges support from NIH
through the Bauer Center for Genomics Research, E.P.W.
acknowledges an NSF postdoctoral fellowship in Biological Informatics, and B.J.E. was supported by an NSF
Faculty Early Career Development Program award and a
Department of Energy Los Alamos National Laboratory
grant. M.E.M. acknowledges support from the Department
of Energy through a Los Alamos National Laboratory–
University of New Mexico Collaborative Research grant.
Supporting Online Material
www.sciencemag.org/cgi/content/full/312/5771/198b/DC1
Methods
References
9 December 2005; accepted 15 March 2006
10.1126/science.1123679
www.sciencemag.org